The Model Evolution Calculus

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The Model Evolution Calculus

• Peter Baumgartner, MPI Saarbruecken and U Koblenz
• Cesare Tinelli, U Iowa

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Background

• Recent research in propositional satisfiability
  (SAT) has been very successful.
• An effective method for SAT was pioneered by
  Davis, Putman, Logemann, and Loveland (DPLL).
• The best modern SAT solvers (MiniSat, zChaff,
  Berkmin,) are based on DPLL.

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The DPLL Procedure Main Idea
assert p T
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The DPLL Procedure Main Idea
assert p T
assert q F
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The DPLL Procedure Main Idea
assert p T
assert q F
guess r T
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The DPLL Procedure Main Idea
assert p T
assert q F
guess r T
contradiction!
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The DPLL Procedure Main Idea
assert p T
assert q F
guess r F
satisfiable!
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Correctness of DPLL method
Prop. A formula ? is satisfiable iff there is a
sequence of guesses such that DPLL(?) ?
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Research Questions

• Can we lift DPLL to the first-order level?
• Can we combine successful SAT techniques (unit
  propagation, backjumping, learning,) with
  successful first-order techniques (unification,
  subsumption, ...)?

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Previous Work

• Instance based methods- (O)SHL Plaisted, -
  Disconnection method Billon, Letz, Stenz,-
  Hyper Tableaux Next Generation Baumgartner,-
  Primal/Dual approach Hooker et al, -
  Ganzinger-Korovin method
• First-Order DPLL Baumgartner- proper lifting
  of split rule

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This Work

• The Model Evolution Calculus
• First-Order DPLL
• DPLLs simplification rules
• Universal variables
• The calculus is a direct lifting of the whole
  DPLL to the first-order level.

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Overview

• The DPLL method as a sequent-style calculus
• A model generation view of DPLL
• The Model Evolution calculus as a lifting of the
  DPLL calculus
• Properties of the ME calculus
• Further Work

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The DPLL Calculus
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The DPLL Calculus
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The DPLL Calculus
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The DPLL Calculus (cont.)
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The DPLL Calculus Key Insight
? can be seen as a finite representation of a
Herbrand interpretation
If I? does not satisfy ?, repair it by adding
literals to ?
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Some Notation
Examples
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The DPLL Calculus RevisitedA Model Evolution
View
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The DPLL Calculus RevisitedA Model Evolution
View
Note
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The DPLL Calculus RevisitedA Model Evolution
View
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Lifting DPLL to First Order Logic

• Main questions
• How to use contexts to represent a FOL Herbrand
  interpretation
• What is a contradictory context
• How to check
• How to check
• How to repair an interpretation

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First-order Contexts

• Sets ? of parametric literals L(u,v,..) and
• universal literals L(x,y,)
• parameters (u,v, ) and variables (x,y,) both
  stand for ground terms
• (roughly) a parametric literal L in ? denotes all
  of its ground instances, unless
  ?L?? for some instance L of L
• a universal literal denotes all of its ground
  instances, unconditionally

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First-order Contexts Examples
? p(u,v)
p(u,v)

• ? produces every instance of p(u,v)

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First-order Contexts Examples
? p(u,v), ?p(u,u)
p(u,v)
?p(u,u)

• ? produces every instance of p(u,v) except the
  instances of p(u,u)
• ? produces every instance of ?p(u,u)

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First-order Contexts Examples
? p(u,v), ?p(u,u), p(f(u),f(u))
p(u,v)
?p(u,u)
p(f(u),f(u))
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First-order Contexts Examples
? ?p(f(u),v), p(u,g(v))
OK
?p(f(u),v)
p(u,g(v))
p(f(u),g(v))
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First-order Contexts Examples
? ?p(f(u),v), p(u,g(v)), p(b,g(v))
OK
?p(f(u),v)
p(u,g(v))
p(b,g(v))
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First-order Contexts Examples
? ?p(u,v), p(u,v)
Not OK! Contradictory
?p(u,v)
p(u,v)
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First-order Contexts Examples
? p(u,v), ?p(x,x)
p(u,v)
?p(x,x)
? produces every instance of ?p(x,x) with no
possible exceptions
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First-order Contexts Examples
? p(u,v), ?p(x,x), p(f(u),f(u)
Not OK! Contradictory
p(u,v)
?p(x,x)
p(f(u),f(u))
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Initial Context
? ?v
?v

• Lambda produces no positive literals

• Well consider only extensions of ?v

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Contexts and Interpretations
Let ? be a non-contradictory context with
parametric literals and universal literals ?
denotes a Herbrand interpretation
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Checking

• ? is called a context unifier (of the clause
  against ?)

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Checking

• Example
• I?p(u,v) ?(p(x,y) ? p(x,x))
• equivalently, match p(x,y), p(x,x) against
  p(,)

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The Model Evolution CalculusSemantical View
Exactly the same as in DPLL!
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The Model Evolution CalculusSemantical View
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The Split Rule Example
First, identify falsified clause instance
Now, split with abs(x)u ?abs(c)u
Clauses xy ? yx abs(x)0
?(xy) ? ?(yz) ? (xz)
Context abs(x)0 vu ?v
?(abs(x)0) ? ?(0u) ? (abs(x)u)
? admissible context unifier
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Example
abs(u)a ? abs(x)a, ...
abs(x)a, abs(u)a ? abs(x)a, ...
abs(x)a ? abs(x)a, ...
abs(x)a ? ?abs(f(x))a ? p(x), ...
abs(x)a ? p(x), ...
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Further Notions

• Derivation tree
• Exhausted/closed branch
• Derivation/refutation
• Limit tree
• Fair limit tree/derivation

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Main Results Completeness
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Main Results Soundness and Completeness
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Main Results Proof Convergence
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Making ME Efficient
Well-known DPLL improvements

• Literal selection strategies Model
  Elimination can exploit dont care
  nondeterminism for remainer literal to split
  on
• Learning (lemma generation) not trivial
  future work
• Intelligent backtracking (backjumping)

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Backjumping
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Backjumping
L not used to close left subtree
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Conclusions

• Full lifting of DPLL achieved
• Properties of DPLL preserved
• sound and complete
• proof convergent
• simplification rules
• model generation paradigm
• (no Commit rule as in FDPLL)
• Abstract framework
• Wide range for fair strategies
• Semantically justified redundancy criteria

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Further Work

• Implement the calculus! (in progress)
• Lift DPLL optimizations (backjumping, lemma
  generation, )
• Add equality
• Study decidable fragments
• Add nonmonotonic features
• Build-in theories
• ...